Theorem sbal1 1341

Description: A theorem used in elimination of disjoint variable restriction on x and y by replacing it with a distinctor -. A.xx = z.

Assertion

Ref	Expression
sbal1	|- (-. A.x x = z -> ([z / y]A.xph <-> A.x[z / y]ph))

Distinct variable group:   x,y

Proof of Theorem sbal1

Step	Hyp	Ref	Expression
1		sbequ12 1177	. . . . 5 |- (y = z -> (A.xph <-> [z / y]A.xph))
2	1	a4s 981	. . . 4 |- (A.y y = z -> (A.xph <-> [z / y]A.xph))
3		sbequ12 1177	. . . . . 6 |- (y = z -> (ph <-> [z / y]ph))
4	3	a4s 981	. . . . 5 |- (A.y y = z -> (ph <-> [z / y]ph))
5	4	dral2 1151	. . . 4 |- (A.y y = z -> (A.xph <-> A.x[z / y]ph))
6	2, 5	bitr3d 528	. . 3 |- (A.y y = z -> ([z / y]A.xph <-> A.x[z / y]ph))
7	6	a1d 12	. 2 |- (A.y y = z -> (-. A.x x = z -> ([z / y]A.xph <-> A.x[z / y]ph)))
8		hba1 1000	. . . . . . 7 |- (A.xph -> A.xA.xph)
9	8	hbsb4 1243	. . . . . 6 |- (-. A.x x = z -> ([z / y]A.xph -> A.x[z / y]A.xph))
10		ax-4 970	. . . . . . . 8 |- (A.xph -> ph)
11	10	sbimi 1169	. . . . . . 7 |- ([z / y]A.xph -> [z / y]ph)
12	11	19.20i 989	. . . . . 6 |- (A.x[z / y]A.xph -> A.x[z / y]ph)
13	9, 12	syl6 22	. . . . 5 |- (-. A.x x = z -> ([z / y]A.xph -> A.x[z / y]ph))
14	13	adantl 388	. . . 4 |- ((-. A.y y = z /\ -. A.x x = z) -> ([z / y]A.xph -> A.x[z / y]ph))
15		sb4 1218	. . . . . . . 8 |- (-. A.y y = z -> ([z / y]ph -> A.y(y = z -> ph)))
16	15	19.20ii 992	. . . . . . 7 |- (A.x -. A.y y = z -> (A.x[z / y]ph -> A.xA.y(y = z -> ph)))
17	16	hbnaes 1144	. . . . . 6 |- (-. A.y y = z -> (A.x[z / y]ph -> A.xA.y(y = z -> ph)))
18		ax-7 959	. . . . . 6 |- (A.xA.y(y = z -> ph) -> A.yA.x(y = z -> ph))
19	17, 18	syl6 22	. . . . 5 |- (-. A.y y = z -> (A.x[z / y]ph -> A.yA.x(y = z -> ph)))
20		ax-16 1206	. . . . . . . . . . 11 |- (A.x x = y -> (y = z -> A.x y = z))
21	20	a1d 12	. . . . . . . . . 10 |- (A.x x = y -> (-. A.x x = z -> (y = z -> A.x y = z)))
22		ax-12 965	. . . . . . . . . 10 |- (-. A.x x = y -> (-. A.x x = z -> (y = z -> A.x y = z)))
23	21, 22	pm2.61i 126	. . . . . . . . 9 |- (-. A.x x = z -> (y = z -> A.x y = z))
24		19.20 991	. . . . . . . . 9 |- (A.x(y = z -> ph) -> (A.x y = z -> A.xph))
25	23, 24	syl9 57	. . . . . . . 8 |- (-. A.x x = z -> (A.x(y = z -> ph) -> (y = z -> A.xph)))
26	25	19.20ii 992	. . . . . . 7 |- (A.y -. A.x x = z -> (A.yA.x(y = z -> ph) -> A.y(y = z -> A.xph)))
27		sb2 1173	. . . . . . 7 |- (A.y(y = z -> A.xph) -> [z / y]A.xph)
28	26, 27	syl6 22	. . . . . 6 |- (A.y -. A.x x = z -> (A.yA.x(y = z -> ph) -> [z / y]A.xph))
29	28	hbnaes 1144	. . . . 5 |- (-. A.x x = z -> (A.yA.x(y = z -> ph) -> [z / y]A.xph))
30	19, 29	sylan9 468	. . . 4 |- ((-. A.y y = z /\ -. A.x x = z) -> (A.x[z / y]ph -> [z / y]A.xph))
31	14, 30	impbid 514	. . 3 |- ((-. A.y y = z /\ -. A.x x = z) -> ([z / y]A.xph <-> A.x[z / y]ph))
32	31	ex 373	. 2 |- (-. A.y y = z -> (-. A.x x = z -> ([z / y]A.xph <-> A.x[z / y]ph)))
33	7, 32	pm2.61i 126	1 |- (-. A.x x = z -> ([z / y]A.xph <-> A.x[z / y]ph))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953  [wsbc 1166

This theorem is referenced by:  sbal 1342

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168

Copyright terms: Public domain